We introduce a one-parameter generalized oscillator algebra A κ (that covers the case of the harmonic oscillator algebra) and discuss its finite-and infinite-dimensional representations according to the sign of the parameter κ. We define an (Hamiltonian) operator associated with A κ and examine the degeneracies of its spectrum. For the finite (when κ < 0) and the infinite (when κ ≥ 0) representations of A κ , we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated generalized oscillator algebra A κ,s , where s denotes the truncation order. We construct two types of temporally stable states for A κ,s (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of A κ,s ). Two applications are considered in this article. The first concerns physical realizations of A κ and A κ,s in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Pöschl-Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.

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